Game Theory

I was analysing the payoff matrix of a game which is quite similar to the Prisoner's Dilemma.

The game is from a game show on TV here called Golden Balls. For those who are not familiar, it is a show which starts with a number of players and in the first few rounds players are eliminated and a cash prize is built up between them. In the final round there are only two players left, and each of them is faced with an option - split or steal...

The rules are as follows:

If both split, then the cash prize is halved between them.
If one splits and one steals, then the splitter gets nothing and the stealer gets it all.
If both steal, then both get nothing.

Clearly the best decision is to steal, as that offers the highest payoff, with no EXTRA risk of losing everything than if you split. However, what is the Nash Equilibria?

I find that all outcomes apart from the one where both split is a Nash Equilibrium. I'm using the definition that a Nash Equilibrium is an outcome such that, if either player changed their decision with the other staying the same, their payoff would not increase. Am I right?

I was analysing the payoff matrix of a game which is quite similar to the Prisoner's Dilemma.

The game is from a game show on TV here called Golden Balls. For those who are not familiar, it is a show which starts with a number of players and in the first few rounds players are eliminated and a cash prize is built up between them. In the final round there are only two players left, and each of them is faced with an option - split or steal...

The rules are as follows:

If both split, then the cash prize is halved between them.
If one splits and one steals, then the splitter gets nothing and the stealer gets it all.
If both steal, then both get nothing.

Clearly the best decision is to steal, as that offers the highest payoff, with no EXTRA risk of losing everything than if you split. However, what is the Nash Equilibria?

I find that all outcomes apart from the one where both split is a Nash Equilibrium. I'm using the definition that a Nash Equilibrium is an outcome such that, if either player changed their decision with the other staying the same, their payoff would not increase. Am I right?

The Nash equilibrium is where both steal.

to be in a Nash equilibrium, each player must answer negatively to the question: "Knowing the strategies of the other players, and treating the strategies of the other players as set in stone, can I benefit by changing my strategy?

Are you sure that's the ONLY nash equilibrium? I know this is the case for the Prisoner's Dilemma, but this differs slightly from that.

Consider the case when player A steals and player B splits. If Player A were to change strategy (such that both split), he would get half instead of all, and if player B chooses to change strategy (such that both steal) then both of them would get neither. Hence both answer the question negatively. Neither can benefit from changing strategy in any of the cases except the split-split outcome.